Quasi-redirecting boundaries of non-positively curved groups

TL;DR

This work generalizes the Gromov boundary through the quasi-redirecting (QR) boundary and proves its existence for several nonpositively curved groups, including Croke–Kleiner admissible groups acting on $ obreak \mathrm{CAT}(0)$ spaces, relatively hyperbolic groups with QR-peripherals, fundamental groups of irreducible non-geometric 3-manifolds, and planar right-angled Coxeter groups. The authors develop a comprehensive framework using templates and quasi-geodesic spirals (backward and forward) to analyze hyperbolic-like directions, establish the necessary QR-Assumptions, and construct the cone-like topology on $X\cup P(X)$ that yields the QR boundary as a topological invariant. They further connect QR boundaries to Bowditch boundaries in the relatively hyperbolic setting and provide structural descriptions of QR boundaries (including non-Hausdorff behavior) for CK-admissible groups, with corollaries for RAAGs on trees. The results open avenues for leveraging QR boundaries in dynamics, random walks, and rigidity, and they offer a robust framework to study boundary phenomena across a broad spectrum of non-hyperbolic groups. The findings have concrete implications for understanding the asymptotic geometry of 3-manifold groups and certain RACGs, enriching the boundary toolbox beyond the classical Gromov boundary and linking to Morse-like directions in non-positive curvature contexts.

Abstract

The quasi-redirecting (QR) boundary is a close generalization of the Gromov boundary to all finitely generated groups. In this paper, we establish that the QR boundary exists as a topological space for several well-studied classes of groups. These include fundamental groups of irreducible non-geometric 3-manifolds, groups that are hyperbolic relative to subgroups with well-defined QR boundaries, right-angled Artin groups whose defining graphs are trees, and right-angled Coxeter groups whose defining flag complexes are planar. This result significantly broadens the known existence of QR boundaries. Additionally, we give a complete characterization of the QR boundaries of Croke-Kleiner admissible groups that act geometrically on CAT(0) spaces. We show that these boundaries are non-Hausdorff and can be understood as one-point compactifications of the Morse-like directions. Finally, we prove that if G is hyperbolic relative to subgroups with well-defined QR boundaries, then the QR boundary of G maps surjectively onto its Bowditch boundary.

Figures (9)

• Figure 1: The ray $\alpha$ can be quasi-redirected to $\beta$ at radius $r$.
• Figure 2: The picture provides a complete description of the poset $P(G)$. The largest element, $[\zeta^*]$, is positioned at the top, while the minimal elements are at the bottom. Both the set of sublinearly Morse elements and the set of non-sublinearly Morse elements have uncountable cardinality.
• Figure 3: Two instances in which $\alpha$ can be quasi-redirected to $\beta$ by $\gamma$. Here, $\alpha$ and $\beta$ are shown as dashed lines, and $\gamma$ is shown as a solid blue line.
• Figure 6: A transition point $(\beta_0)_r$ separates the point $p$ and any geodesic line that connects $\xi(\textbf{b})$ and $\xi(\textbf{a})$.
• Figure 7: The figure illustrates a portion of a backward spiral path in a right-angled template

Theorems & Definitions (95)

• Definition 1.1
• Theorem A
• Definition 1.2
• Theorem B
• Corollary C
• proof
• Theorem D
• Definition 1.3
• Corollary E: Theorem \ref{['maincox']}
• Definition 2.1: Quasi-Geodesics